Some Fixed Point Theorems on Ordered Metric Spaces and Application
© I. Altun and H. Simsek. 2010
Received: 2 July 2009
Accepted: 13 January 2010
Published: 21 January 2010
We present some fixed point results for nondecreasing and weakly increasing operators in a partially ordered metric space using implicit relations. Also we give an existence theorem for common solution of two integral equations.
Existence of fixed points in partially ordered sets has been considered recently in , and some generalizations of the result of  are given in [2–6]. Also, in  some applications to matrix equations are presented, in [3, 4] some applications to periodic boundary value problem and to some particular problems are, respectively, given. Later, in  O'Regan and Petruşel gave some existence results for Fredholm and Volterra type integral equations. In some of the above works, the fixed point results are given for nondecreasing mappings.
We can order the purposes of the paper as follows.
First, we give a slight generalization of some of the results of the above papers using an implicit relation in the following way.
Afterwards, in , the authors used the nonlinear contractive condition, that is,
where is anondecreasing function with for , instead of (1.1). Also in , the authors proved a fixed point theorem using generalized nonlinear contractive condition, that is,
In some of the above mentioned theorems, the fixed point results are given for nondecreasing mappings. Also in these theorems the following condition is used:
In Section 4, we give some examples such that two weakly increasing mappings need not be nondecreasing. Therefore, we give a common fixed point theorem for two weakly increasing operators in partially ordered metric spaces using implicit relation technique. Also we did not use the condition (1.6) in this theorem. At the end, to see the applicability of our result, we give an existence theorem for common solution of two integral equations using a result of the Section 4.
2. Implicit Relation
3. Fixed Point Theorem for Nondecreasing Mappings
We need the following lemma for the proof of our theorems.
Lemma 3.1 (See ).
Note that if we take that
If we combine Theorem 3.2 with Example 2.1, we obtain the following result.
Theorem of  follows from Example 2.3, Remark 3.3, and Theorem 3.2.
We can have some new results from other examples and Theorem 3.2.
and hence has a unique fixed point. If condition (3.27) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in .
4. Fixed Point Theorem for Weakly Increasing Mappings
Now we give a fixed point theorem for two weakly increasing mappings in ordered metric spaces using an implicit relation. Before this, we will define an implicit relation for the contractive condition of the theorem.
Note that, two weakly increasing mappings need not be nondecreasing.
Proof of Theorem 4.5.
For example, we can have the following corollary.
Consider the integral equations
Consider the integral equations (5.1).
Then the integral equations (5.1) have a unique common solution in .
Then is a partially ordered set. Also is a complete metric space. Moreover, for any increasing sequence in converging to , we have for any . Also for every , there exists which is comparable to and .
The authors thank the referees for their appreciation, valuable comments, and suggestions.
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